Before You Begin

Question #41537

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$ . A planar region $S$ in $\mathbb{R}^3$ consists of all points whose position vector $\vec{r}$ satisfies $\vec{r} \cdot (\vec{a} \times \vec{b}) = 0$ , $|\vec{r} \times \vec{a}| \le 6\sqrt{3}$ , and $|\vec{r} \times \vec{b}| \le 9\sqrt{3}$ . If $R$ is the orthogonal projection of the region $S$ onto the $xy$ -plane, find the value of the integral $\iint_R (x - y)^2 \,dx\,dy$ .

Options

A.
104
B.
156
C.
312
D.
936

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Axis of Integration Selection|Geometric Interpretation of Cross Product|Collinearity and Coplanarity|Line lying in a plane

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of the cross product and the transition from 3D vector spaces to 2D multiple integration.

The primary trap lies in assuming standard Cartesian boundaries or ignoring the Jacobian determinant when changing the basis.

Students often incorrectly assume that the bounds on the parameters $\alpha$ and $\beta$ can be directly applied to the $x$ and $y$ axes, leading to the trap answer of 104.

Furthermore, it exploits the oversight where students fail to realize that a region lying in a plane defined by non-orthogonal vectors necessitates a linear transformation to correctly evaluate the projected area integral.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Axis of Integration Selection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free